Original Solution: JavaScript - 261 255 228 227 179 153 Characters
/(\d)(\1(\d|.{6}|.{9})|(\d|.{6}|.{9})\1|.{7}\1(.|.{9})|(.|.{9})\1.{7}|(.{7,9}|.{17})\1.{8}|.{8}\1(.{7,9}|.{17}))\1/.test(s.replace(/\n/g,'A'))?'yes':'no'
Assuming that the string to test is in the variable s (to make it a function f then add f=s=> to the beginning of the code or, otherwise, to take input from a prompt then replace s with prompt()).
Outputs is to the console.
3rd Solution: JavaScript (ECMAScript 6) - 178 Characters
p=x=>parseInt(x,36);for(t="2313ab1b8a2a78188h9haj9j8iaiir9r",i=v=0;s[i];i++)for(j=0;t[j];v|=s[i]==s[i+a]&s[i]==s[i+b]&i%9<8&(b>3|(i+b-a)%9<8))a=p(t[j++]),b=p(t[j++]);v?'yes':'no'
I took the 2nd solution, below, (which uses regular expressions to check for characters in certain configurations) and reworked it to just check the string for identical characters in the same configurations without using regular expressions.
The Base-36 string "2313ab1b8a2a78188h9haj9j8iaiir9r" gives pairs of offsets to check - i.e. the pair 23 results in the check if ith character is identical to the (i+2)th character and the (i+3)th character (the equivalent of the regular expression (.).\1\1 - with some additional checks to ensure that the non-identical character is not a newline).
2nd Solution: JavaScript (ECMAScript 6) - 204 Characters
p=x=>parseInt(x,18);g=a=>a?a>1?"(.|\\n){"+a+"}":".":"";f=(x,a,b)=>RegExp("(.)"+g(a)+"\\1"+g(b)+"\\1").test(x);for(t="10907160789879h8",i=v=0;t[i];v|=f(s,x,y)||f(s,y,x))x=p(t[i++]),y=p(t[i++]);v?'yes':'no'
Builds multiple regular expressions (see below for more details) using pairs of values taken from the Base-18 string 10907160789879h8 and takes the OR of all the tests. To reduce it further you can note that the regular expressions come in pairs where one is the "reverse" of the other (ignoring the Regular Expressions for 3-in-a-row horizontally and vertically as the OP states they will never be present - if you want to add those tests back in the append 0088 to the Base-18 string).
Explanation
Start with 16 regular expressions covering all the possible configurations of valid moves:
REs=[
/(\d)\1\1/, // 3-in-a-row horizontally
/(\d).\1\1/, // 3-in-a-row horizontally after left-most shifts right
/(\d)\1.\1/, // 3-in-a-row horizontally after right-most shifts left
/(\d)(?:.|\n){9}\1\1/, // 3-in-a-row horizontally after left-most shifts down
/(\d)(?:.|\n){7}\1.\1/, // 3-in-a-row horizontally after middle shifts down
/(\d)(?:.|\n){6}\1\1/, // 3-in-a-row horizontally after right-most shifts down
/(\d)\1(?:.|\n){6}\1/, // 3-in-a-row horizontally after left-most shifts up
/(\d).\1(?:.|\n){7}\1/, // 3-in-a-row horizontally after middle shifts up
/(\d)\1(?:.|\n){9}\1/, // 3-in-a-row horizontally after right-most shifts up
/(\d)(?:.|\n){7,9}\1(?:.|\n){8}\1/, // 3-in-a-row vertically (with optional top shifting left or right)
/(\d)(?:.|\n){7}\1(?:.|\n){9}\1/, // 3-in-a-row vertically after middle shifts right
/(\d)(?:.|\n){9}\1(?:.|\n){7}\1/, // 3-in-a-row vertically after middle shifts left
/(\d)(?:.|\n){8}\1(?:.|\n){7}\1/, // 3-in-a-row vertically after bottom shifts right
/(\d)(?:.|\n){8}\1(?:.|\n){9}\1/, // 3-in-a-row vertically after bottom shifts left
/(\d)(?:.|\n){17}\1(?:.|\n){8}\1/, // 3-in-a-row vertically after top shifts down
/(\d)(?:.|\n){8}\1(?:.|\n){17}\1/, // 3-in-a-row vertically after bottom shifts up
];
(Note: the regexs for 3-in-a-row horizontally (0th) and vertically (part of the 9th) are irrelevant as the OP states that inputs matching these will never be present.)
Testing each of those against the input will determine if a valid move of that configuration can be found.
However, the regular expressions can be combined to give these 6:
/(\d)(?:.|(?:.|\n){9}|(?:.|\n){6})?\1\1/ // Tests 0,1,3,5
/(\d)\1(?:.|(?:.|\n){9}|(?:.|\n){6})?\1/ // Tests 0,2,6,8
/(\d)(?:.|\n){7}\1(?:.|(?:.|\n){9})\1/ // Tests 4,10
/(\d)(?:.|(?:.|\n){9})\1(?:.|\n){7}\1/ // Tests 7,11
/(\d)(?:(?:.|\n){7,9}|(?:.|\n){17})\1(?:.|\n){8}\1/ // Tests 9,14
/(\d)(?:.|\n){8}\1(?:(?:.|\n){7,9}|(?:.|\n){17})\1/ // Tests 9a,12,13,15
These can then be combined into a single regular expression:
/(\d)(?:.|(?:.|\n){9}|(?:.|\n){6})?\1\1|(\d)\2(?:.|(?:.|\n){9}|(?:.|\n){6})?\2|(\d)(?:.|\n){7}\3(?:.|(?:.|\n){9})\3|(\d)(?:.|(?:.|\n){9})\4(?:.|\n){7}\4|(\d)(?:(?:.|\n){7,9}|(?:.|\n){17})\5(?:.|\n){8}\5|(\d)(?:.|\n){8}\6(?:(?:.|\n){7,9}|(?:.|\n){17})\6/
Which just needs to be tested against the input.
Test Cases
Some test cases which other people might find useful (doesn't comply with the input format of using only digits 1-7 but that's easily corrected and is only an 8x4 grid - since that is the minimum required for a test of all the valid inputs).
In the format of a map from input string to which of the 16 regular expressions above it matches.
Tests={
"12345678\n34567812\n56781234\n78123456": -1, // No Match
"12345678\n34969912\n56781234\n78123456": 1, // 3-in-a-row horizontally after left-most shifts right
"12345678\n34567812\n59989234\n78123456": 2, // 3-in-a-row horizontally after right-most shifts left
"12345978\n34567899\n56781234\n78123456": 3, // 3-in-a-row horizontally after left-most shifts down
"12345978\n34569892\n56781234\n78123456": 4, // 3-in-a-row horizontally after middle shifts down
"12345678\n34967812\n99781234\n78123456": 5, // 3-in-a-row horizontally after right-most shifts down
"12399678\n34967812\n56781234\n78123456": 6, // 3-in-a-row horizontally after left-most shifts up
"12345678\n34597912\n56789234\n78123456": 7, // 3-in-a-row horizontally after middle shifts up
"12345998\n34567819\n56781234\n78123456": 8, // 3-in-a-row horizontally after right-most shifts up
"12945678\n34597812\n56791234\n78123456": 9, // 3-in-a-row vertically after top shifts right
"12349678\n34597812\n56791234\n78123456": 9, // 3-in-a-row vertically after top shifts left
"12345978\n34569812\n56781934\n78123456": 10, // 3-in-a-row vertically after middle shifts right
"92345678\n39567812\n96781234\n78123456": 11, // 3-in-a-row vertically after middle shifts left
"12945678\n34967812\n59781234\n78123456": 12, // 3-in-a-row vertically after bottom shifts right
"12349678\n34569812\n56781934\n78123456": 13, // 3-in-a-row vertically after bottom shifts left
"12395678\n34567812\n56791234\n78193456": 14, // 3-in-a-row vertically after top shifts down
"12345698\n34567892\n56781234\n78123496": 15, // 3-in-a-row vertically after bottom shifts up
"12345678\n34567899\n96781234\n78123456": -1, // No match - Matches (.)\1.\1 but not 3 in a row
"12345679\n99567812\n56781234\n78123456": -1, // No match - Matches (.).\1\1 but not 3 in a row
};
Edit 1
Replace \ds with . - saves 6 characters.
Edit 2
Replace (?:.|\n) with [\s\S] and removed extra non-capturing groups and updated back references (as suggested by m-buettner) and added in yes/no output.
Edit 3
- Added ECMAScript 6 solution to build the individual Regular Expressions from a Base-18 string.
- Removed the tests for 3-in-a-row horizontally (as suggested by m-buettner).
Edit 4
Added another (shorter) solution and two more non-matching tests cases.
Edit 5
- Shortened original solution by replacing newlines with a non-numeric character (as suggested by VadimR).
Edit 6
- Shortened original solution by combining bits of the regular expression (as suggested by VadimR).
